Homological invariants for classification of kinematic singularities

Abstract This paper provides a classification of kinematic singularities for the mechanisms, serial manipulators and arrays of control moment gyros via homological invariants. It is shown here that rather than solely using local geometric methods to describe kinematic singularity type, analysis of subsets of the input space through their associated homological invariants provide the additional ability to classify and identify specific types of kinematic singularities with nonlocal behaviour such as those of the ”degenerate” type (path-connected impassable singularities) and those that are only locally passable. In addition, as a bi-product of the analysis, a partition of the input space by its fibres of a smooth map over points in the workspace is given that separates regions of singular from nonsingular points. Some proofs are provided utilizing singular and cellular homology on the existence of singularity types and classes within these fibres of the input space.

[1]  Jochen Pfalzgraf,et al.  On geometric and topological reasoning in robotics , 1997, Annals of Mathematics and Artificial Intelligence.

[2]  Valeri N. Platonov,et al.  Singular Surfaces in Unsupported Gyrodyne Systems , 1978 .

[3]  Tsuneo Yoshikawa,et al.  Manipulability of Robotic Mechanisms , 1985 .

[4]  Zexiang Li,et al.  Singularities of parallel manipulators: a geometric treatment , 2003, IEEE Trans. Robotics Autom..

[5]  Clément Gosselin,et al.  Singularity analysis of closed-loop kinematic chains , 1990, IEEE Trans. Robotics Autom..

[6]  F. Park,et al.  Singularity Analysis of Closed Kinematic Chains , 1999 .

[7]  Joseph A. Paradiso,et al.  Steering law design for redundant single-gimbal control moment gyroscopes. [for spacecraft attitude control] , 1990 .

[8]  Allen R. Tannenbaum,et al.  Robotic manipulators and the geometry of real semialgebraic sets , 1987, IEEE J. Robotics Autom..

[9]  Dinesh K. Pai,et al.  Genericity and singularities of robot manipulators , 1992, IEEE Trans. Robotics Autom..

[10]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[11]  Nazareth Bedrossian,et al.  Classification of singular configurations for redundant manipulators , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[12]  Peter Donelan.  Kinematic Singularities of Robot Manipulators , 2010 .

[13]  Bong Wie,et al.  Singularity Analysis and Visualization for Single-Gimbal Control Moment Gyro Systems , 2003 .

[14]  Y. Matsuoka,et al.  Robotics for surgery. , 1999, Annual review of biomedical engineering.

[15]  Haruhisa Kurokawa.  Survey of Theory and Steering Laws of Single-Gimbal Control Moment Gyros , 2007 .

[16]  R. Ho.  Algebraic Topology , 2022 .

[17]  Dimiter Stefanov Zlatanov,et al.  Generalized singularity analysis of mechanisms , 1998 .

[18]  Peter Donelan.  SINGULARITIES OF ROBOT MANIPULATORS , 2007 .